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We have studied the iso-height lines on the $\mathrm{WO_3}$ surface as aphysical candidate for conformally invariant curves. We have shown that theselines are conformally invariant with the same statistics of domain walls in thecritical Ising model. They belong to the family of conformal invariant curvescalled Schramm-Loewner evolution (or $SLE_{\kappa}$), with diffusivity of$\kappa \sim 3$. This can be regarded as the first experimental observation ofSLE curves. We have also argued that Ballistic Deposition (BD) can serve as agrowth model giving rise to contours with similar statistics at large scales.

Conformal Curves on $WO_3$ Surface